Hyperbolic analogues of fullerenes with face-types (6,9) and (6,10)

Abstract

Mathematical models of fullerenes are cubic polyhedral and spherical maps of face-type (5, 6), that is, with pentagonal and hexagonal faces only. Any such map necessarily contains exactly 12 pentagons, and it is known that for any integer alpha >= 0 except alpha = 1 there exists a fullerene map with precisely alpha hexagons. In this paper we consider hyperbolic analogues of fullerenes, modelled by cubic polyhedral maps of face-type (6, k), where k is an element of {9, 10}, on orientable surface of genus at least two. The number of k-gons in this case depends on the genus but the number of hexagons is again independent of the surface. For every triple k is an element of {9, 10}, g >= 2 and alpha >= 0, we determine if there exists a cubic polyhedral map of face-type (6, k) with exactly alpha hexagons on an orientable surface of genus g. The only unsolved cases are k = 10, g = 5 and alpha <= 3 when we are not able to say if a hyperbolic fullerene with these parameters exists.